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   <title>Tutorial on Convolutional Coding with Viterbi Decoding--Description of the Data Generation, Convolutional Encoding, Channel Mapping and AWGN, and Quantizing Algorithms</title>
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<a NAME="algorithms"></a><b><font face="Arial"><font size=+1>Description
of the Algorithms&nbsp; (Part 1)</font></font></b>
<p>&nbsp;The steps involved in simulating a communication channel using
convolutional encoding and Viterbi decoding are as follows:
<ul>
<li>
<a href="#genalgorithm">Generate the data</a> to be transmitted through
the channel-result is binary data bits</li>

<li>
<a href="#conalgorithm">Convolutionally encode</a> the data-result is channel
symbols</li>

<li>
<a href="#mapping">Map the one/zero channel symbols</a> onto an antipodal
baseband signal, producing transmitted channel symbols</li>

<li>
<a href="#addnoise">Add noise</a> to the transmitted channel symbols-result
is received channel symbols</li>

<li>
<a href="#quantizing">Quantize</a> the received channel levels-one bit
quantization is called hard-decision, and two to n bit quantization is
called soft-decision (n is usually three or four)</li>

<li>
<a href="algrthms2.html">Perform Viterbi decoding</a> on the quantized
received channel symbols-result is again binary data bits</li>

<li>
Compare the decoded data bits to the transmitted data bits and count the
number of errors.</li>
</ul>
<i>Many of you will notice that I left out the steps of modulating the
channel symbols onto a transmitted carrier, and then demodulating the received
carrier to recover the channel symbols. You're right, but we can accurately
model the effects of AWGN even though we bypass those steps.</i>
<p><a NAME="genalgorithm"></a><b><i><font face="Arial">Generating the Data</font></i></b>
<p>Generating the data to be transmitted through the channel can be accomplished
quite simply by using a random number generator. One that produces a uniform
distribution of numbers on the interval 0 to a maximum value is provided
in C: <tt>rand ()</tt>. Using this function, we can say that any value
less than half of the maximum value is a zero; any value greater than or
equal to half of the maximum value is a one.
<p><a NAME="conalgorithm"></a><b><i><font face="Arial">Convolutionally
Encoding the Data</font></i></b>
<p>Convolutionally encoding the data is accomplished using a shift register
and associated combinatorial logic that performs modulo-two addition. (A
shift register is merely a chain of flip-flops wherein the output of the
nth flip-flop is tied to the input of the (n+1)th flip-flop. Every time
the active edge of the clock occurs, the input to the flip-flop is clocked
through to the output, and thus the data are shifted over one stage.) The
combinatorial logic is often in the form of cascaded exclusive-or gates.
As a reminder, exclusive-or gates are two-input, one-output gates often
represented by the logic symbol shown below,
<center>
<p><img SRC="figs/xor_gate.gif" ALT="exclusive-or gate symbol" height=64 width=93></center>

<p>that implement the following truth-table:
<br>&nbsp;
<br>&nbsp;
<center><table BORDER CELLPADDING=7 WIDTH="218" >
<tr>
<td VALIGN=TOP WIDTH="28%">
<center><b><tt>Input A</tt></b></center>
</td>

<td VALIGN=TOP WIDTH="27%">
<center><b><tt>Input B</tt></b></center>
</td>

<td VALIGN=TOP WIDTH="45%">
<center><b><tt>Output</tt></b>
<p><b><tt>(A xor B)</tt></b></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="28%">
<center><tt>0</tt></center>
</td>

<td VALIGN=TOP WIDTH="27%">
<center><tt>0</tt></center>
</td>

<td VALIGN=TOP WIDTH="45%">
<center><tt>0</tt></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="28%">
<center><tt>0</tt></center>
</td>

<td VALIGN=TOP WIDTH="27%">
<center><tt>1</tt></center>
</td>

<td VALIGN=TOP WIDTH="45%">
<center><tt>1</tt></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="28%">
<center><tt>1</tt></center>
</td>

<td VALIGN=TOP WIDTH="27%">
<center><tt>0</tt></center>
</td>

<td VALIGN=TOP WIDTH="45%">
<center><tt>1</tt></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="28%">
<center><tt>1</tt></center>
</td>

<td VALIGN=TOP WIDTH="27%">
<center><tt>1</tt></center>
</td>

<td VALIGN=TOP WIDTH="45%">
<center><tt>0</tt></center>
</td>
</tr>
</table></center>

<p>The exclusive-or gate performs modulo-two addition of its inputs. When
you cascade q two-input exclusive-or gates, with the output of the first
one feeding one of the inputs of the second one, the output of the second
one feeding one of the inputs of the third one, etc., the output of the
last one in the chain is the modulo-two sum of the q + 1 inputs.
<p>Another way to illustrate the modulo-two adder, and the way that is
most commonly used in textbooks, is as a circle with a + symbol inside,
thus:
<center>
<p><img SRC="figs/ringsum.gif" ALT="modulo-two adder symbol" height=48 width=48></center>

<p>Now that we have the two basic components of the convolutional encoder
(flip-flops comprising the shift register and exclusive-or gates comprising
the associated modulo-two adders) defined, let's look at a picture of a
convolutional encoder for a rate 1/2, K = 3, m = 2 code:
<br>&nbsp;
<br>&nbsp;
<br>
<center>
<p><img SRC="figs/ce_7_5_a.gif" ALT="rate 1/2 K = 3 (7, 5) convolutional encoder" height=232 width=600></center>

<p>In this encoder, data bits are provided at a rate of k bits per second.
Channel symbols are output at a rate of n = 2k symbols per second. The
input bit is stable during the encoder cycle. The encoder cycle starts
when an input clock edge occurs. When the input clock edge occurs, the
output of the left-hand flip-flop is clocked into the right-hand flip-flop,
the previous input bit is clocked into the left-hand flip-flop, and a new
input bit becomes available. Then the outputs of the upper and lower modulo-two
adders become stable. The output selector (SEL A/B block) cycles through
two states-in the first state, it selects and outputs the output of the
upper modulo-two adder; in the second state, it selects and outputs the
output of the lower modulo-two adder.
<p>The encoder shown above encodes the K = 3, (7, 5) convolutional code.
The octal numbers 7 and 5 represent the code generator polynomials, which
when read in binary (111<sub>2</sub> and 101<sub>2</sub>) correspond to
the shift register connections to the upper and lower modulo-two adders,
respectively. This code has been determined to be the "best" code for rate
1/2, K = 3. It is the code I will use for the remaining discussion and
examples, for reasons that will become readily apparent when we get into
the Viterbi decoder algorithm.
<p>Let's look at an example input data stream, and the corresponding output
data stream:
<p>Let the input sequence be 010111001010001<sub>2</sub>.
<p>Assume that the outputs of both of the flip-flops in the shift register
are initially cleared, i.e. their outputs are zeroes. The first clock cycle
makes the first input bit, a zero, available to the encoder. The flip-flop
outputs are both zeroes. The inputs to the modulo-two adders are all zeroes,
so the output of the encoder is 00<sub>2</sub>.
<p>The second clock cycle makes the second input bit available to the encoder.
The left-hand flip-flop clocks in the previous bit, which was a zero, and
the right-hand flip-flop clocks in the zero output by the left-hand flip-flop.
The inputs to the top modulo-two adder are 100<sub>2</sub>, so the output
is a one. The inputs to the bottom modulo-two adder are 10<sub>2</sub>,
so the output is also a one. So the encoder outputs 11<sub>2</sub> for
the channel symbols.
<p>The third clock cycle makes the third input bit, a zero, available to
the encoder. The left-hand flip-flop clocks in the previous bit, which
was a one, and the right-hand flip-flop clocks in the zero from two bit-times
ago. The inputs to the top modulo-two adder are 010<sub>2</sub>, so the
output is a one. The inputs to the bottom modulo-two adder are 00<sub>2</sub>,
so the output is zero. So the encoder outputs 10<sub>2</sub> for the channel
symbols.
<p>And so on. The timing diagram shown below illustrates the process:
<br>&nbsp;
<br>&nbsp;
<br>
<center>
<p><img SRC="figs/ce_td.gif" ALT="timing diagram for rate 1/2 convolutional encoder" height=322 width=600></center>

<p><br>
<br>
<br>
<p>After all of the inputs have been presented to the encoder, the output
sequence will be:
<p>00 11 10 00 01 10 01 11 11 10 00 10 11 00 11<sub>2</sub>.
<p>Notice that I have paired the encoder outputs-the first bit in each
pair is the output of the upper modulo-two adder; the second bit in each
pair is the output of the lower modulo-two adder.
<p>You can see from the structure of the rate 1/2 K = 3 convolutional encoder
and from the example given above that each input bit has an effect on three
successive pairs of output symbols. That is an extremely important point
and that is what gives the convolutional code its error-correcting power.
The reason why will become evident when we get into the Viterbi decoder
algorithm.
<p>Now if we are only going to send the 15 data bits given above, in order
for the last bit to affect three pairs of output symbols, we need to output
two more pairs of symbols. This is accomplished in our example encoder
by clocking the convolutional encoder flip-flops two ( = m) more times,
while holding the input at zero. This is called "flushing" the encoder,
and results in two more pairs of output symbols. The final binary output
of the encoder is thus 00 11 10 00 01 10 01 11 11 10 00 10 11 00 11 10
11<sub>2</sub>. If we don't perform the flushing operation, the last m
bits of the message have less error-correction capability than the first
through (m - 1)th bits had. This is a pretty important thing to remember
if you're going to use this FEC technique in a burst-mode environment.
So's the step of clearing the shift register at the beginning of each burst.
The encoder must start in a known state and end in a known state for the
decoder to be able to reconstruct the input data sequence properly.
<p>Now, let's look at the encoder from another perspective. You can think
of the encoder as a simple state machine. The example encoder has two bits
of memory, so there are four possible states. Let's give the left-hand
flip-flop a binary weight of 2<sup>1</sup>, and the right-hand flip-flop
a binary weight of 2<sup>0</sup>. Initially, the encoder is in the all-zeroes
state. If the first input bit is a zero, the encoder stays in the all zeroes
state at the next clock edge. But if the input bit is a one, the encoder
transitions to the 10<sub>2</sub> state at the next clock edge. Then, if
the next input bit is zero, the encoder transitions to the 01<sub>2</sub>
state, otherwise, it transitions to the 11<sub>2</sub> state. The following
table gives the next state given the current state and the input, with
the states given in binary:
<br>&nbsp;
<br>&nbsp;
<center><table BORDER CELLSPACING=2 CELLPADDING=7 WIDTH="282" >
<tr>
<td VALIGN=TOP WIDTH="33%"><font face="Arial"><font size=-1>&nbsp;</font></font></td>

<td VALIGN=TOP COLSPAN="2" WIDTH="67%">
<center><a NAME="statetable"></a><b><font face="Arial"><font size=-1>Next
State, if&nbsp;</font></font></b></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><b><font face="Arial"><font size=-1>Current State</font></font></b></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><b><font face="Arial"><font size=-1>Input = 0:</font></font></b></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><b><font face="Arial"><font size=-1>Input = 1:</font></font></b></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>00</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>00</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>10</font></font></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>01</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>00</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>10</font></font></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>10</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>01</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>11</font></font></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>11</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>01</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>11</font></font></center>
</td>
</tr>
</table></center>

<br>&nbsp;
<p>The above table is often called a state transition table. We'll refer
to it as the <tt>next state</tt> table.<tt> </tt>Now let us look at a table
that lists the channel output symbols, given the current state and the
input data, which we'll refer to as the <tt>output</tt> table:
<br>&nbsp;
<br>&nbsp;
<center><table BORDER CELLSPACING=2 CELLPADDING=7 WIDTH="282" >
<tr>
<td VALIGN=TOP WIDTH="33%"></td>

<td VALIGN=TOP COLSPAN="2" WIDTH="67%">
<center><a NAME="outputtable"></a><b><font face="Arial"><font size=-1>Output
Symbols, if</font></font></b></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><b><font face="Arial"><font size=-1>Current State</font></font></b></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><b><font face="Arial"><font size=-1>Input = 0:</font></font></b></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><b><font face="Arial"><font size=-1>Input = 1:</font></font></b></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>00</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>00</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>11</font></font></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>01</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>11</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>00</font></font></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>10</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>10</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>01</font></font></center>
</td>
</tr>

<tr>
<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>11</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>01</font></font></center>
</td>

<td VALIGN=TOP WIDTH="33%">
<center><font face="Arial"><font size=-1>10</font></font></center>
</td>
</tr>
</table></center>

<br>&nbsp;
<p>You should now see that with these two tables, you can completely describe
the behavior of the example rate 1/2, K = 3 convolutional encoder. Note
that both of these tables have 2<sup>(K - 1)</sup> rows, and 2<sup>k</sup>
columns, where K is the constraint length and k is the number of bits input
to the encoder for each cycle. These two tables will come in handy when
we start discussing the Viterbi decoder algorithm.
<p><a NAME="mapping"></a><b><i><font face="Arial">Mapping the Channel Symbols
to Signal Levels</font></i></b>
<p>Mapping the one/zero output of the convolutional encoder onto an antipodal
baseband signaling scheme is simply a matter of translating zeroes to +1s
and ones to -1s. This can be accomplished by performing the operation y
= 1 - 2x on each convolutional encoder output symbol.
<p><a NAME="addnoise"></a><b><i><font face="Arial">Adding Noise to the
Transmitted Symbols</font></i></b>
<p>Adding noise to the transmitted channel symbols produced by the convolutional
encoder involves generating Gaussian random numbers, scaling the numbers
according to the desired energy per symbol to noise density ratio, E<sub>s</sub>/N<sub>0</sub>,
and adding the scaled Gaussian random numbers to the channel symbol values.
<p>For the uncoded channel, E<sub>s</sub>/N<sub>0 </sub>= E<sub>b</sub>/N<sub>0</sub>,
since there is one channel symbol per bit.&nbsp; However, for the coded
channel, E<sub>s</sub>/N<sub>0 </sub>= E<sub>b</sub>/N<sub>0</sub> + 10log<sub>10</sub>(k/n).&nbsp;
For example, for rate 1/2 coding, E<sub>s</sub>/N<sub>0 </sub>= E<sub>b</sub>/N<sub>0</sub>
+ 10log<sub>10</sub>(1/2) = E<sub>b</sub>/N<sub>0</sub> - 3.01 dB.&nbsp;
Similarly, for rate 2/3 coding, E<sub>s</sub>/N<sub>0 </sub>= E<sub>b</sub>/N<sub>0</sub>
+ 10log<sub>10</sub>(2/3) = E<sub>b</sub>/N<sub>0</sub> - 1.76 dB.
<p>The Gaussian random number generator is the only interesting part of
this task. C only provides a uniform random number generator, <tt>rand()</tt>.
In order to obtain Gaussian random numbers, we take advantage of relationships
between uniform, Rayleigh, and Gaussian distributions:
<p>Given a uniform random variable U, a Rayleigh random variable R can
be obtained by:
<p><img SRC="figs/eqn01.gif" ALT="equation for Rayleigh random deviate given uniform random deviate" height=30 width=297 align=ABSCENTER>
<p>where&nbsp;<img SRC="figs/eqn02.gif" height=24 width=24 align=ABSCENTER>is
the variance of the Rayleigh random variable, and given R and a second
uniform random variable V, two Gaussian random variables G and H can be
obtained by
<p><i>G</i> = <i>R</i> cos <i>U</i> and <i>H</i> = <i>R</i> sin <i>V</i>.
<p>In the AWGN channel, the signal is corrupted by additive noise, n(t),
which has the power spectrum <i>No</i>/2 watts/Hz. The variance&nbsp;<img SRC="figs/eqn02.gif" ALT="variance" height=24 width=24 align=ABSBOTTOM>of
this noise is equal to&nbsp;<img SRC="figs/eqn03.gif" ALT="noise density div by two" height=22 width=38 align=TEXTTOP>.
If we set the energy per symbol <i>E<sub>s</sub></i> equal to 1, then&nbsp;<img SRC="figs/eqn04.gif" ALT="equation relating variance to SNR" height=28 width=110 align=ABSBOTTOM>.
So&nbsp;<img SRC="figs/eqn05.gif" ALT="equation for AWGN st dev given SNR" height=28 width=139 align=ABSCENTER>.
<p><a NAME="quantizing"></a><b><i><font face="Arial">Quantizing the Received
Channel Symbols</font></i></b>
<p>An ideal Viterbi decoder would work with infinite precision, or at least
with floating-point numbers. In practical systems, we quantize the received
channel symbols with one or a few bits of precision in order to reduce
the complexity of the Viterbi decoder, not to mention the circuits that
precede it. If the received channel symbols are quantized to one-bit precision
(&lt; 0V = 1, <u>></u> 0V = 0), the result is called hard-decision data.
If the received channel symbols are quantized with more than one bit of
precision, the result is called soft-decision data. A Viterbi decoder with
soft decision data inputs quantized to three or four bits of precision
can perform about 2 dB better than one working with hard-decision inputs.
The usual quantization precision is three bits. More bits provide little
additional improvement.
<p>The selection of the quantizing levels is an important design decision
because it can have a significant effect on the performance of the link.
The following is a very brief explanation of one way to set those levels.
Let's assume our received signal levels in the absence of noise are -1V
= 1, +1V = 0. With noise, our received signal has mean +/- 1 and standard
deviation&nbsp;<img SRC="figs/eqn05.gif" ALT="equation for AWGN st dev given SNR" height=28 width=139 align=ABSCENTER>.
Let's use a uniform, three-bit quantizer having the input/output relationship
shown in the figure below, where D is a decision level that we will calculate
shortly:
<center>
<p><img SRC="figs/quantize.gif" ALT="8-level quantizer function plot" height=342 width=384></center>

<p>The decision level, D, can be calculated according to the formula&nbsp;<img SRC="figs/eqn06.gif" ALT="equation for quantizer decision level" height=28 width=228 align=ABSCENTER>,
where E<sub>s</sub>/N<sub>0</sub> is the energy per symbol to noise density
ratio<i>. (The above figure was redrawn from Figure 2 of Advanced Hardware
Architecture's ANRS07-0795, "Soft Decision Thresholds and Effects on Viterbi
Performance". See the </i><a href="fecbiblio.html">bibliography</a><i>&nbsp;
for a link to their web pages.)</i>
<p>Click <a href="algrthms2.html">here</a> to proceed to the description
of the Viterbi decoding algorithm itself...
<p>Or click on one of the links below to go to the beginning of that section:
<p>&nbsp;<a href="tutorial.html">Introduction</a>
<br>&nbsp;<a href="algrthms2.html">Description of the Algorithms&nbsp;
(Part 2)</a>
<br>&nbsp;<a href="examples.html">Simulation Source Code Examples</a>
<br>&nbsp;<a href="simrslts.html">Example Simulation Results</a>
<br>&nbsp;<a href="fecbiblio.html">Bibliography</a>
<br>&nbsp;<a href="tutorial.html#specapps">About Spectrum Applications...</a>
<br>&nbsp;
<br>&nbsp;
<br>
<br>
<center>
<p><img SRC="figs/stripe.gif" height=6 width=600></center>

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